Finite Group Databasebeta

← C3×#29701 #10648308

Subgroups — C3×#29701 #10648308

Order 768.

940 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#541C211yesmaximal, characteristic
2C2×C3×#479C212yesmaximal, complemented
3C3×#2629C211yesmaximal, characteristic
4C3×#2059C212yesmaximal, complemented
5C3×#2043C212yesmaximal, complemented
6C3×#2036C212yesmaximal
7C3×#2035C212yesmaximal
8C3×#1803C212yesmaximal, complemented
9C3×#1798C211yesmaximal, characteristic
10#29701C311yesmaximal, characteristic, direct product factor, Sylow
11C3×#541—28no—
12C2×C3×C4⋊Q8C2211yescharacteristic
13C2×C3×#126—22no—
14C2×C3×#125—22no—
15C2×C3×#125C2211yescharacteristic
16C2×C3×#125C2211yescharacteristic
17C2×C3×C22⋊Q8C2212yes—
18C2×C3×C4×Q8C2212yes—
19C2×C3×#117C2211yescharacteristic
20C3×#479—28no—
21C2×C3×#107C2212yes—
22C2×C3×#106C2212yes—
23C2×C3×C4⋊C8C2212yes—
24C2×C3×#103C2212yes—
25C2×C3×#103C2212yes—
26C2×C3×#103C2212yes—
27C2×C3×#103C2212yes—
28C2×C3×C22⋊C8C2211yescharacteristic
29C2×C3×C22⋊C8C2211yescharacteristic
30C3×#398—22no—
31C3×#396—22no—
32C3×#389C2212yes—
33C3×#384C2211yescharacteristic
34C3×#380C2212yes—
35C3×#340C2212yes—
36C3×#340C2212yes—
37C3×#336C2212yes—
38C2×#541C2×C311yescharacteristic
39C2×#479C2×C312yescomplemented
40#2629C2×C311yescharacteristic
41#2059C2×C312yescomplemented
42#2043C2×C312yescomplemented
43#2036C2×C312yes—
44#2035C2×C312yes—
45#1803C2×C312yescomplemented
46#1798C2×C311yescharacteristic
47C22×C3×Q8C2312yes—
48C23×C3×C4C2311yescharacteristic
49C22×C3×C8C2312yes—
50C22×C3×C8C2312yes—
51C3×C4⋊Q8—24no—
52C3×#126—216no—
53C3×#125—216no—
54C3×#125—24no—
55C3×#125—24no—
56C3×C22⋊Q8—24no—
57C3×C22⋊Q8—24no—
58C3×C4×Q8—24no—
59C3×C4×Q8D418yes—
60C3×#117—24no—
61C2×C3×C4⋊C4—22no—
62C2×C3×C4⋊C4—22no—
63C2×C3×C4⋊C4—22no—
64C2×C3×C4⋊C4—22no—
65C2×C3×C4⋊C4—22no—
66C2×C3×C4⋊C4—22no—
67C2×C3×C4⋊C4—22no—
68C2×C3×C4⋊C4D412yes—
69C2×C3×C4⋊C4C2312yes—
70C2×C3×C4⋊C4C2312yes—
71C2×C3×C4⋊C4C2312yes—
72C2×C3×C22⋊C4—22no—
73C2×C3×C22⋊C4—22no—
74C2×C3×C22⋊C4D412yes—
75C2×C3×C42—22no—
76C2×C3×C42C2311yescharacteristic
77C2×C3×C42C2311yescharacteristic
78C3×#107—24no—
79C3×#106—24no—
80C3×C4⋊C8—24no—
81C3×#103—24no—
82C3×#103—24no—
83C3×#103—42no—
84C3×#103—42no—
85C3×C22⋊C8—41no—
86C3×C22⋊C8—41no—
87C3×#95—22no—
88C3×#95—22no—
89#541—28no—
90C2×C4⋊Q8C22×C311yescharacteristic
91C2×#126—22no—
92C2×#125—22no—
93C2×#125C22×C311yescharacteristic
94C2×#125C22×C311yescharacteristic
95C2×C22⋊Q8C22×C312yes—
96C2×C4×Q8C22×C312yes—
97C2×#117C22×C311yescharacteristic
98#479—28no—
99C2×#107C22×C312yes—
100C2×#106C22×C312yes—
101C2×C4⋊C8C22×C312yes—
102C2×#103C22×C312yes—
103C2×#103C22×C312yes—
104C2×#103C22×C312yes—
105C2×#103C22×C312yes—
106C2×C22⋊C8C22×C311yescharacteristic
107C2×C22⋊C8C22×C311yescharacteristic
108#398—22no—
109#396—22no—
110#389C22×C312yes—
111#384C22×C311yescharacteristic
112#380C22×C312yes—
113#340C22×C312yes—
114#340C22×C312yes—
115#336C22×C312yes—
116C24×C3C2×D411yescharacteristic
117C2×C3×Q8—22no—
118C2×C3×Q8—22no—
119C2×C3×Q8—22no—
120C2×C3×Q8—22no—
121C2×C3×Q8—22no—
122C2×C3×Q8C2×D414yes—
123C22×C3×C4—22no—
124C22×C3×C4—22no—
125C22×C3×C4—22no—
126C22×C3×C4—21no—
127C22×C3×C4—22no—
128C22×C3×C4—21no—
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D412yes—
133C22×C3×C4C2×D412yes—
134C22×C3×C4C2411yescharacteristic
135C2×C3×C8—22no—
136C2×C3×C8—22no—
137C2×C3×C8—22no—
138C2×C3×C8—22no—
139C2×C3×C8—22no—
140C2×C3×C8—22no—
141C3×C4⋊C4—28no—
142C3×C4⋊C4—28no—
143C3×C4⋊C4—28no—
144C3×C4⋊C4—28no—
145C3×C4⋊C4—24no—
146C3×C4⋊C4—22no—
147C3×C4⋊C4—22no—
148C3×C4⋊C4—22no—
149C3×C4⋊C4—22no—
150C3×C4⋊C4—22no—
151C3×C4⋊C4—44no—
152C3×C4⋊C4—82no—
153C3×C4⋊C4—82no—
154C3×C4⋊C4C2×D414yes—
155C3×C22⋊C4—28no—
156C3×C22⋊C4—24no—
157C3×C22⋊C4—82no—
158C3×C42—28no—
159C3×C42—22no—
160C3×C42C2×D414yes—
161C22×Q8C23×C312yes—
162C23×C4C23×C311yescharacteristic
163C22×C8C23×C312yes—
164C22×C8C23×C312yes—
165C4⋊Q8—24no—
166#126—216no—
167#125—216no—
168#125—24no—
169#125—24no—
170C22⋊Q8—24no—
171C22⋊Q8—24no—
172C4×Q8—24no—
173C4×Q8C3×D418yes—
174#117—24no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4C3×D412yes—
183C2×C4⋊C4C23×C312yes—
184C2×C4⋊C4C23×C312yes—
185C2×C4⋊C4C23×C312yes—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4C3×D412yes—
189C2×C42—22no—
190C2×C42C23×C311yescharacteristic
191C2×C42C23×C311yescharacteristic
192#107—24no—
193#106—24no—
194C4⋊C8—24no—
195#103—24no—
196#103—24no—
197#103—42no—
198#103—42no—
199C22⋊C8—41no—
200C22⋊C8—41no—
201#95—22no—
202#95—22no—
203C23×C3—21no—
204C23×C3—21no—
205C23×C3—21no—
206C23×C3—22no—
207C23×C3—22no—
208C23×C3C22×D411yescharacteristic
209C3×Q8—24no—
210C3×Q8—24no—
211C3×Q8—42no—
212C3×Q8—42no—
213C2×C3×C4—22no—
214C2×C3×C4—22no—
215C2×C3×C4—22no—
216C2×C3×C4—24no—
217C2×C3×C4—24no—
218C2×C3×C4—21no—
219C2×C3×C4—21no—
220C2×C3×C4—21no—
221C2×C3×C4—22no—
222C2×C3×C4—22no—
223C2×C3×C4—42no—
224C2×C3×C4—42no—
225C2×C3×C4—42no—
226C2×C3×C4—42no—
227C2×C3×C4—42no—
228C2×C3×C4—42no—
229C2×C3×C4—42no—
230C2×C3×C4—42no—
231C2×C3×C4—42no—
232C2×C3×C4—42no—
233C2×C3×C4—42no—
234C2×C3×C4—42no—
235C2×C3×C4#12014yes—
236C2×C3×C4C22×D412yes—
237C2×C3×C4H(5,𝔽2)11yescharacteristic
238C2×C3×C42−1+411yescharacteristic
239C2×C3×C42−1+412yes—
240C3×C8—42no—
241C3×C8—42no—
242C24C2×C3×D411yescharacteristic
243C2×Q8—22no—
244C2×Q8—22no—
245C2×Q8—22no—
246C2×Q8—22no—
247C2×Q8—22no—
248C2×Q8C2×C3×D414yes—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—21no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4C2×C3×D411yescharacteristic
258C22×C4C2×C3×D412yes—
259C22×C4C2×C3×D412yes—
260C22×C4C24×C311yescharacteristic
261C2×C8—22no—
262C2×C8—22no—
263C2×C8—22no—
264C2×C8—22no—
265C2×C8—22no—
266C2×C8—22no—
267C4⋊C4—28no—
268C4⋊C4—28no—
269C4⋊C4—28no—
270C4⋊C4—28no—
271C4⋊C4—24no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—44no—
278C4⋊C4—82no—
279C4⋊C4—82no—
280C4⋊C4C2×C3×D414yes—
281C22⋊C4—28no—
282C22⋊C4—24no—
283C22⋊C4—82no—
284C42—28no—
285C42—22no—
286C42C2×C3×D414yes—
287C22×C3—41no—
288C22×C3—42no—
289C22×C3—41no—
290C22×C3—41no—
291C22×C3—42no—
292C22×C3C2×#12011yescharacteristic
293C22×C3#53611yescharacteristic
294C22×C3#53711yescharacteristic
295C22×C3#57712yes—
296C22×C3#57812yes—
297C3×C4—24no—
298C3×C4—41no—
299C3×C4—44no—
300C3×C4—42no—
301C3×C4—44no—
302C3×C4—82no—
303C3×C4—82no—
304C3×C4#53612yes—
305C3×C4#53712yes—
306C23—22no—
307C23—22no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23C22×C3×D411yescharacteristic
312Q8—24no—
313Q8—24no—
314Q8—42no—
315Q8—42no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4C3×#12014yes—
339C2×C4C22×C3×D412yes—
340C2×C4C3×H(5,𝔽2)11yescharacteristic
341C2×C4C3×2−1+411yescharacteristic
342C2×C4C3×2−1+412yes—
343C8—42no—
344C8—42no—
345C2×C3—81no—
346C2×C3#261611yescharacteristic
347C2×C3#301312yes—
348C2×C3#324212yes—
349C2×C3#326011yescharacteristic
350C2×C3#326111yescharacteristic
351C22—41no—
352C22—42no—
353C22—41no—
354C22—41no—
355C22—42no—
356C22C2×C3×#12011yescharacteristic
357C22C3×#53611yescharacteristic
358C22C3×#53711yescharacteristic
359C22C3×#57712yes—
360C22C3×#57812yes—
361C4—24no—
362C4—41no—
363C4—44no—
364C4—42no—
365C4—44no—
366C4—82no—
367C4—82no—
368C4C3×#53612yes—
369C4C3×#53712yes—
370C3#2970111yescharacteristic, direct product factor, Sylow
371C2—81no—
372C2C3×#261611yescharacteristic
373C2C3×#301312yes—
374C2C3×#324212yes—
375C2C3×#326011yescharacteristic
376C2C3×#326111yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC24×C3C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×C3×#479C2semidirect
4. C3×#2059C2semidirect
5. C3×#2043C2semidirect
8. C3×#1803C2semidirect
10. #29701C3direct (N × H)
39. C2×#479C2×C3semidirect
41. #2059C2×C3semidirect
42. #2043C2×C3semidirect
45. #1803C2×C3semidirect
370. C3#29701direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×C3×#5412
C2×C3×#4792
C3×#26292
C3×#20592
C3×#20432
C3×#20362
C3×#20352
C3×#18032
C3×#17982
#297013